Galaxies are always in motion relative to the Milky Way. My question is, which galaxy is receding the fastest from our viewpoint?
What is the theorised mechanism that causes this?
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This question may be a duplicate to: On what scale does the universe expand?
The mechanism for the expansion of the universe is gravity, manifested through what's known as Dark Energy. Dark Energy is a component to the universe which today, composes 68.3% of the energy density of the universe. You may be confused as to how gravity could be responsible for the expansion of the universe, but the equation of state of DE,
w = pressure/density
is negative (w_DE = -1), which means that it has anti-gravitational properties. Additionally, its density remains constant as the coordinates of the universe change, which is the reason why it dominates later on in the universe's history (while all other components, matter (dark or otherwise) and radiation/relativistic matter, become diluted with the expansion of the universe).
To answer your question, because the expansion of the universe is happening at all points in space, it looks like the "center" of expansion is at every point in space (see: Space expansion in layman terms), the galaxies furthest away (highest redshift), are the ones moving quickest away from the Milky Way Galaxy.
The galaxy that is receding away from us fastest is whichever galaxy currently holds the record for the highest measured redshift.
This galaxy obviously changes as more and more discoveries are made. This web page that describes redshift, may keep an up-to-date record of this. It lists UDFy-38135539, discovered in the Hubble Ultra Deep Field, as a previous record holder (see below), with a spectroscopically confirmed redshift of $z=8.5549 \pm 0.0002$. That is, lines in the spectrum appear at wavelengths that are a factor of $(1+z)=9.55$ times longer (redder). But there is apparent controversy over the claimed detection of spectral features in this galaxy. (e.g. Bunker et al. 2013).
Bramnmer et al. (2013) report the "tentative detection" of a possible Lyman alpha emission line at $z=12.12$ for UDFj-39546284, another ultra deep field galaxy.
Either way it seems only a matter of time before $z>9$ galaxies are confirmed, since there are many candidates that have so far been selected only by virtue of their peculiar colours (Lyman break galaxies at near infrared wavelengths, including MACS0647-JD mentioned in comments to other posts). e.g. Bouwens et al. (2014).
Update Oesch et al. (2016) report HST grism spectroscopy of a high redshift galaxy candidate, GN-z11, identified in the CANDELS/GOODS survey. They confirm the presence of a Lyman break indicating a spectroscopic redshift of 11.1. This appears to be the current record holder.
The reasons that these galaxies are receding from us is simply the expansion of the universe. Galaxies that are further away appear to recede faster. The galaxies discussed above must be $>13$ billion light years away according to the current "concordance" cosmological model.
Translating a redshift to a recession velocity requires knowledge of the cosmological parameters. Fig.2 of Davis & Lineweaver (2004) shows the relationship. For our current best estimates of cosmological parameters, a redshift of 11 corresponds to a recession velocity of 2.3c. That is, this galaxy is currently receding from us at 2.3 times the speed of light.